Decision trees are a useful way to describe alternative scenarios and select the decision that maximizes the NPV or whatever the decision maker is trying to optimize. In decision theory, the most desired outcome is based on the goals and preferences of the decision maker. The reservoir engineer can use decision trees to describe complex scenarios with multiple decisions and multiple probabilities. This discussion can only be considered a brief introduction. In constructing a decision tree we use rectangles to represent decision nodes and circles to represent probability nodes. Two or more decisions can be associated with each decision nodes and multiple nodes can be associated with a probability node. A probability node representing betting one thousand dollars on number “30” at a roulette wheel[1] in Las Vegas looks like this:

The single bet on number 30 can easily be evaluated as to its expected value as follows:

In other words, a single $1,000 bet on number 30 (or any other number) has a negative expected value of 52.63 dollars. Similar analyses will show negative expectations for each of the gambling games explaining the fabulous hotels and inexpensive “all you can eat” buffets in Las Vegas. But is it crazy to play roulette or make other decisions selecting lower expected values than other alternatives? No, the decider may have a different use for $35,000 than $1,000. Maybe he owes a debt that it is immediately due and has a major negative result if he is unable to generate $35,000 right away. This particular preference for risk is actually unusual; most people have less utility for expected outcomes that have large negative impacts. This analysis does not mean that every player will lose money playing roulette. It is a relatively straightforward exercise to model a roulette wheel with various strategies in which a significant fraction of the players win[2]. It is the aggregate EMV of all players over the long run that is negative.

Suppose someone gives you the chance to play a game in which a fair coin is flipped. In the case of heads[3], you receive $2.00 and for tails you get nothing. You will no doubt be happy to play this game as it has an expected monetary value (EMV) of $1.00. How much would you be willing sell your ticket for? It is unlikely anyone will pay you much more than $1.00 and if you sell it for much less you are “giving away” EMV. Now consider another game. In this game you have to buy a ticket. In this game a heads pays $3.00 and a tail pays $1.00. How much would you be willing to pay for this ticket? The EMV of this game is $2.00 and if you pay any less than that you are (on an expected value basis) gaining money. Would you pay more than $1.00? If you paid $1.00, the second game becomes equivalent to the first with the net result of a head being $3.00 – $1.00 = $2.00 and the result of a tail would be $1 -$1 = 0. Is there a difference in how much you are willing to sell your ticket for in the first game and what you are willing to pay for it in the second game? Decision makers often make decisions on other than an expected value basis based on how much investment exposure is necessary.

Let’s consider another set of decisions. In the first option, you pay €1,000 by investing in a very small percentage (0.1%) of a drilling well that you anticipate has a 50% chance of success (or a coin flip for heads if you prefer). In the case of a discovery you win a series of cash flows with an NPV of €4,000 while a dry hole pays nothing. The EMV is 0.5*€4,000 – €1,000 = €1,000. Are you interested in this investment? If you believe these numbers and have 1,000 euros (€) to invest it is an obvious decision to participate in the project. Now let’s look at the 100% working interest position. In this case you need to invest €1,000,000 and have a 50% chance of €4,000,000. Assume that your net worth is just enough that you could come up with the money by mortgaging your house, cashing in your retirement and borrowing all of the money that you can; it is unlikely that you would accept such an investment opportunity. A single investment or a series of investments that have the potential to bankrupt an investor is known as “gambler’s ruin.” Your utility for a positive €1,000,000 is considerably less than 1,000 times greater than it is for €1,000. By analyzing your responses to a series of similarly constructed alternatives, an individual with game theory expertise could construct your “indifference curve.” Your personal utility and indifference curves and those of the decision maker are not as important as is the utility functions of the corporation. For our purposes we will assume that the corporation has a unit slope linear utility function and makes its decisions entirely on EMV. Exceptions to this would only occur for massive investments.

In the drill vs farmout example, we had a decision tree that looked like the following:

There were only two decisions, drill or farmout. The probability nodes were only dry hole or discovery. The analysis of a decision tree proceeds from right to left as the EMV is calculated for each probability node. The expected value of each probability node is replaced with its expected value and the highest EMV decision node is selected. There can be multiple probabilities at each probability and the probability node can be replaced by Monte Carlo simulations. In fact, the entire decision tree can be replaced by Monte Carlo simulations with a distribution of decisions being made and the corresponding variability in results conveyed to decision makers.

[1] American style roulette wheels have 38 equally probable outcomes. A bet placed on a single number pays 35$ for each dollar bet if successful. There are many alternative bets such as betting on black or red which would pay 1$ for each dollar bet. In both success cases you keep your bet. As there are two green numbers, 18 black and 18 red, the EMV of betting $1,000 on red or black is -1000 + (18/38)*2000 = -52.63$. European style roulette wheels have only one green slot and 37 possible outcomes with slightly different payoffs that are nonetheless EMV negative for participants.

[2] Consider the trivial case of 38 players each betting $1,000 on one each of the 38 spots and play one time. One person will walk away with $36,000 (his bet plus his $35,000 in winnings) while 37 players lose their bets. The house makes $2,000.

[3] Coin collectors refer to the obverse and reverse of a coin rather than “heads” or “tails” but we will use the more common convention.

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